Partial Derivatives Estimation of Multivariate Variance Function in Heteroscedastic Model via Wavelet Method

Abstract

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems.

1. Introduction

This paper considers the following heteroscedastic model:

$$\begin{array}{c}\hfill Y_i=g\left({\mathit{X}}_{\mathit{i}}\right)+f\left({\mathit{X}}_{\mathit{i}}\right)U_i,i\in \left\{1,\cdots ,n\right\}.\end{array}$$

(1)

In this model, $({\mathit{X}}_{\mathit{1}},Y_1),\cdots ,({\mathit{X}}_{\mathit{n}},Y_n)$ are independent and identically distributed random vectors. The functions $g\left(\mathit{x}\right)$ and $f\left(\mathit{x}\right)$ all are defined on ${[a,b]}^d$. $U_1,\cdots ,U_n$ are identically distributed random variables, which satisfy $\mathbb{E}\left[U_i\right]=0$ and $Var\left[U_i\right]=1$. Furthermore, the random vector ${\mathit{X}}_{\mathit{i}}$ and random variable $U_i$ are uncorrelated for any $i\in \left\{1,\cdots ,n\right\}$. This paper is devoted to estimating the partial derivatives $\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)$ of the variance function $r\left(\mathit{x}\right)(r\left(\mathit{x}\right):=f^2\left(\mathit{x}\right))$ from the observed data $({\mathit{X}}_{\mathit{1}},Y_1),\cdots ,({\mathit{X}}_{\mathit{n}},Y_n)$. The partial derivative $\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)$ is defined by

$$\begin{array}{c}\hfill \left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)=\left({\displaystyle \frac{{\partial }^{\mathit{m}}r}{\partial x_1^{m_1}\cdots \partial x_d^{m_d}}}\right)(x_1,\cdots ,x_d),\end{array}$$

with $\mathit{m}=(m_1,\cdots ,m_d)\in {\mathbb{N}}^d$, and $\left|\mathit{m}\right|={\displaystyle \sum _{i=1}^d}{m}_i$.

In practical applications, the heteroscedastic model is widely used for monitoring the signal-to-noise ratios in quality control [1,2], measuring the reliability of time series prediction [3], evaluating the volatility or risk in financial markets [4] and so on. Hence, many significant results have been obtained by [5,6,7,8] and others. For this heteroscedastic model, Fan and Yao [9] propose a residual-based estimator of the variance function, and study the asymptotic normality properties of an estimator. A class of difference-based kernel estimators of a variance function are constructed by [10]. Moreover, the asymptotic rates of convergence and the optimal bandwidth of kernel estimators are discussed. Wang et al. [11] consider the minimax convergence rates of a kernel estimator over pointwise squared error and global integrated mean squared error, respectively. The optimal estimation of a variance function with random design is discussed by [12]. Zaoui [13] studies the variance function estimation with model selection aggregation and convex aggregation.

Derivative estimation plays a crucial role in nonparametric statistics estimation, big data processing, and other practical applications. For example, some companies predict the profit growth rate when the research and development investment increases. Many financial and fund institutions evaluate the volatility of stock market prices and so on. Many important and interesting results of derivative estimation are obtained using different methods. Zhou and Wolfe [14] propose a spline derivative estimator, and discuss the asymptotic normality and variance property. A spatially adaptive derivative estimator is constructed by [15]. Chaubey et al. [16] consider the upper bound over $L^p$-loss for wavelet estimators of density derivative functions. A convergence rate over the mean integrated error of a derivative estimator for mixing sequences is proved by [17]. For the estimation problem (1), the derivatives estimation of the variance function via the wavelet method is proved by [18]. However, it should be pointed out that those above results all focus on the estimation of the derivatives of a one-dimensional function. There is a lack of partial derivatives estimation of a multivariate variance function. Hence, in this paper, we construct a partial derivatives estimator using the wavelet method, and discuss the convergence rates of the wavelet estimator under different mild conditions.

The structure of this paper is given as follows. Section 2 specifies some mild hypotheses for the estimation model (1), and constructs a wavelet estimator of the partial derivative function. Two important auxiliary results of the wavelet estimator are proved in Section 3. The estimation errors of the wavelet estimator under different assumptions are discussed in Section 4.

2. Wavelet Estimator

In this section, we will give some hypotheses of the estimation problem (1), and construct a partial derivatives estimator using the wavelet method. For the estimation model (1), the following mild assumptions are proposed, which are used in the later discussion.

A1: For the partial derivative $\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)$ with $\mathit{m}=(m_1,\cdots ,m_d)\in {\mathbb{N}}^d$, if any $\mathit{\theta }=({\theta }_1,\cdots ,{\theta }_d)\in {\mathbb{N}}^d$ and ${\theta }_i\le m_i$, the partial derivative function satisfies $\left({\partial }^{\mathit{\theta }}r\right)\left(\mathit{x}\right)=0$ when $x_i\equiv a$ or b.

A2: The function $g\left(\mathit{x}\right)$ is known and bounded, i.e., there exists a positive constant $c_1$ such that $\left|g\left(\mathit{x}\right)\right|\le c_1$.

A3: The density function $h\left(\mathit{x}\right)$ of the random vector $\mathit{X}$ satisfies that $c_2\le h\left(\mathit{x}\right)\le c_3$, where $\mathit{x}\in {[a,b]}^d$, $c_2$ and $c_3$ are two positive constants.

A4: The random variables $Y_1,Y_2,\dots ,Y_n$ are defined on $[c_4,c_5]$, where $c_4$ and $c_5$ are two constants.

In order to construct a wavelet estimator, some basic theories of wavelets are given in the following [19,20]. Let $\Phi$ be a scaling function and $\Psi$ be a wavelet function such that

$$\left\{{\Phi }_{\tau ,\mathit{k}},{\Psi }_{j,\mathit{k},u};j\ge \tau ,u\in \left\{1,\cdots ,2^d-1\right\},\mathit{k}\in {\mathbb{Z}}^d\right\}$$

constitutes an orthonormal basis of $L^2\left({\mathbb{R}}^d\right)$, where $\tau$ is a positive integer,

$$\begin{array}{c}{\Phi }_{j,\mathit{k}}\left(\mathit{x}\right)=2^{\frac{jd}{2}}\Phi (2^j{x}_1-k_1,\cdots ,2^j{x}_d-k_d),\\ {\Psi }_{j,\mathit{k},u}\left(x\right)=2^{\frac{jd}{2}}{\Psi }_u(2^j{x}_1-k_1,\cdots ,2^j{x}_d-k_d).\end{array}$$

Then, for any integer $j_0$ such that $j_0\ge \tau$, a function $f\left(\mathit{x}\right)\in L^2\left({[a,b]}^d\right)$ can be expanded into a wavelet series as:

$$f\left(\mathit{x}\right)=\sum _{\mathit{k}\in {\Lambda }_{j_0}}{\alpha }_{j_0,\mathit{k}}{\Phi }_{j_0,\mathit{k}}\left(\mathit{x}\right)+\sum _{j=j_0}^{\infty }\sum _{u=1}^{2^d-1}\sum _{\mathit{k}\in {\Lambda }_j}{\beta }_{j,\mathit{k},u}{\Psi }_{j,\mathit{k},u}\left(\mathit{x}\right),$$

(2)

where ${\alpha }_{j_0,\mathit{k}}={\int }_{{[a,b]}^d}f\left(\mathit{x}\right)·{\Phi }_{j_0,\mathit{k}}\left(\mathit{x}\right)d\mathit{x}$, ${\beta }_{j,\mathit{k},u}={\int }_{{[a,b]}^d}f\left(\mathit{x}\right)·{\Psi }_{j,\mathit{k},u}\left(\mathit{x}\right)d\mathit{x}$ and the cardinality of ${\Lambda }_j$ satisfies $|{\Lambda }_j|\asymp 2^{jd}$. In this paper, we choose some compactly supported wavelets, such as the Daubechies wavelet [21]. In addition, this paper adopts the following symbol: $A\lesssim B$ denotes $A\le cB$ for some constant $c>0$; $A\gtrsim B$ means $B\lesssim A$; and $A\asymp B$ stand for both $A\lesssim B$ and $B\lesssim A$. For any $\mathit{x}\in {\mathbb{R}}^d$, $\parallel \mathit{x}\parallel :={\displaystyle \sum _{i=1}^d}\left|x_i\right|$.

Now we define a wavelet estimator of partial derivatives function $\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)$ by

$$\begin{array}{c}\hfill \widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right):=\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}{\widehat{\alpha }}_{j_{\ast },\mathit{k}}{\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right).\end{array}$$

(3)

In this definition,

$$\begin{array}{c}\hfill {\widehat{\alpha }}_{j_{\ast },\mathit{k}}:=\frac{{(-1)}^{\left|\mathit{m}\right|}}{n}\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{i}}\right)-{(-1)}^{\left|\mathit{m}\right|}{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}\end{array}$$

(4)

and

$$\begin{array}{cc}\hfill \left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)& =\left({\displaystyle \frac{{\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}}{\partial x_1^{m_1}\cdots \partial x_d^{m_d}}}\right)(x_1,\cdots ,x_d)\hfill \\ \hfill & =2^{\frac{j_{\ast }d}{2}}·2^{j_{\ast }\left|\mathit{m}\right|}\left({\partial }^{\mathit{m}}\Phi \right)(2^{j_{\ast }}{x}_1-k_1,\cdots ,2^{j_{\ast }}{x}_d-k_d).\hfill \end{array}$$

3. Two Lemmas

This section will provide two important lemmas, which are used to prove the main theorem in a later section. According to the following lemma, it is easy to see that our wavelet estimator $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ is unbiased.

Lemma 1.

For the model (1) with A1,

$$\begin{array}{c}\hfill \mathbb{E}\left[{\widehat{\alpha }}_{j_{\ast },\mathit{k}}\right]={\alpha }_{j_{\ast },\mathit{k}}.\end{array}$$

Proof. 

By the definition of ${\widehat{\alpha }}_{j_{\ast },\mathit{k}}$ and the properties of random vectors $({\mathit{X}}_{\mathit{i}},Y_i)$,

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\widehat{\alpha }}_{j_{\ast },\mathit{k}}\right]& =\mathbb{E}\left[\frac{{(-1)}^{\left|\mathit{m}\right|}}{n}\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{i}}\right)-{(-1)}^{\left|\mathit{m}\right|}{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}\right]\hfill \\ \hfill & =\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{Y_1^2}{h\left({\mathit{X}}_{\mathit{1}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]-{(-1)}^{\left|\mathit{m}\right|}{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}.\hfill \end{array}$$

Then, it follows from (1) that

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\widehat{\alpha }}_{j_{\ast },\mathit{k}}\right]\hfill \\ \hfill & =\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{r\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}{U}_1^2\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]+2\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{f\left({\mathit{X}}_{\mathit{1}}\right)g\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}{U}_1\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]\hfill \\ \hfill & +\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{g^2\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]-{(-1)}^{\left|\mathit{m}\right|}{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}.\hfill \end{array}$$

Note that the conditions $\mathbb{E}\left[U_1\right]=0$ and $Var\left[U_1\right]=1$ imply $\mathbb{E}\left[U_1^2\right]=1$. Furthermore, using the assumption of no correlation between $U_i$ and $X_i$, one gets

$$\begin{array}{c}\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{r\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}{U}_1^2\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]=\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{r\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]\end{array}$$

and

$$\begin{array}{c}\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{f\left({\mathit{X}}_{\mathit{1}}\right)g\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}{U}_1\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]=0.\end{array}$$

In addition, because the density function of the random vector $\mathit{X}$ is $h\left(\mathit{x}\right)$, the following equation can be obtained easily:

$$\begin{array}{c}\hfill \mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{g^2\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right)\right]={(-1)}^{\left|\mathit{m}\right|}{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}.\end{array}$$

According to the above results, one has

$$\begin{array}{c}\hfill \mathbb{E}\left[{\widehat{\alpha }}_{j_{\ast },\mathit{k}}\right]=\mathbb{E}\left[{(-1)}^{\left|\mathit{m}\right|}\frac{r\left({\mathit{X}}_{\mathit{1}}\right)}{h\left({\mathit{X}}_{\mathit{1}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{1}}\right).\right]\end{array}$$

Then, it is easy to see from A1 that

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\widehat{\alpha }}_{j_{\ast },\mathit{k}}\right]& ={(-1)}^{\left|\mathit{m}\right|}{\int }_{{[a,b]}^d}r\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}\hfill \\ \hfill & ={\int }_{{[a,b]}^d}\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right){\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)d\mathit{x}={\alpha }_{j_{\ast },\mathit{k}}.\hfill \end{array}$$

□

For nonparametric estimation, wavelet estimators can be viewed as generalized kernel estimators. For any $\mathit{u},\mathit{v}\in {\mathbb{R}}^d$, we introduce the kernel $K(\mathit{u},\mathit{v})$ by $K(\mathit{u},\mathit{v})={\displaystyle \sum _{\mathit{k}\in {\mathbb{Z}}^d}}\Phi (\mathit{u}-\mathit{k})\Phi (\mathit{v}-\mathit{k}).$ Now, we give some important properties of this kernel function, which will be used in the later discussion. Furthermore, we define

$$\begin{array}{c}\hfill K^{\left(\mathit{m}\right)}(\mathit{u},\mathit{v}):=\sum _{\mathit{k}\in {\mathbb{Z}}^d}\Phi (\mathit{u}-\mathit{k})\left({\partial }_{\mathit{v}}^{\mathit{m}}\Phi \right)(\mathit{v}-\mathit{k}),\end{array}$$

where $K^{\left(\mathit{m}\right)}(\mathit{u},\mathit{v}):=\left({\partial }_{\mathit{v}}^{\mathit{m}}K\right)(\mathit{u},\mathit{v})$ denotes the $\mathit{m}$th partial derivative of $K(\mathit{u},\mathit{v})$ with respect to $\mathit{v}$.

Let the scaling function $\Phi$ be $\lambda$-regular [20,22,23], i.e., $\Phi \in C^{\lambda }$ and $|D^{\alpha }\Phi \left(\mathit{x}\right)|\le c_l{(1+\parallel \mathit{x}\parallel )}^{-l}$ for any integer $l\ge 1$, $\mathit{\alpha }=({\alpha }_1,\cdots ,{\alpha }_d)\in {\mathbb{N}}^d$ with $\left|\mathit{\alpha }\right|={\displaystyle \sum _{i=1}^d}{\alpha }_i\le \lambda$ and $\mathit{x}\in {\mathbb{R}}^d$. Then, there exists a positive constant $C_d$ such that

$$\begin{array}{c}\hfill |K^{\left(\mathit{m}\right)}(\mathit{u},\mathit{v})|\le {\displaystyle \frac{C_d}{{(1+\parallel \mathit{u}-\mathit{v}\parallel )}^{d+1}}}.\end{array}$$

(5)

Meanwhile, one can obtain that

$$\begin{array}{c}\hfill |K^{\left(\mathit{m}\right)}(\mathit{u},\mathit{y})-K^{\left(\mathit{m}\right)}(\mathit{v},\mathit{y})|\lesssim \parallel \mathit{u}-\mathit{v}\parallel .\end{array}$$

(6)

For more properties and details of kernel functions, one can see [24,25].

Lemma 2.

For the model (1) with conditions A3 and A4, the wavelet estimator $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ is defined by (3) and $2^{j_{\ast }}\lesssim {\left(\frac{n}{lnn}\right)}^{1/d}$, there exists a constant $\kappa >0$ such that

$$\begin{array}{c}\hfill \mathbb{P}\left[\left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\right|\ge \kappa {\eta }_n\right]\lesssim n^{-z\left(\kappa \right)},\end{array}$$

where $z\left(\kappa \right)=\frac{{\kappa }^2}{2(1+\kappa /3)}$ and ${\eta }_n\asymp 2^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\sqrt{\frac{lnn}{n}}$.

Proof. 

According to the definition of $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ by (3),

$$\begin{array}{cc}\hfill & \left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\right|\hfill \\ \hfill & =\left|\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}({\widehat{\alpha }}_{j_{\ast },\mathit{k}}-\mathbb{E}\left[{\widehat{\alpha }}_{j_{\ast },\mathit{k}}\right]){\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)\right|\hfill \\ \hfill & =\frac{1}{n}\left|\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}\left(\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{i}}\right)-\mathbb{E}\left[\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{i}}\right)\right]\right){\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)\right|\hfill \\ \hfill & =\frac{1}{n}\left|\sum _{i=1}^n\left(\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})-\mathbb{E}\left[\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right]\right)\right|=\frac{1}{n}\left|\sum _{i=1}^n{B}_i\right|,\hfill \end{array}$$

where $B_i:=\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})-\mathbb{E}\left[\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right]$ and

$$\begin{array}{cc}\hfill K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})& :=\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}{\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{i}}\right)\hfill \\ \hfill & =\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}{2}^{\frac{j_{\ast }d}{2}}\Phi (2^{j_{\ast }}{x}_1-k_1,\cdots ,2^{j_{\ast }}{x}_d-k_d)·\hfill \\ & 2^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\left({\partial }^{\mathit{m}}\Phi \right)(2^{j_{\ast }}{X}_{i_1}-k_1,\cdots ,2^{j_{\ast }}{X}_{i_d}-k_d)\hfill \\ \hfill & =2^{j_{\ast }(d+|\mathit{m}\left|\right)}{K}^{\left(\mathit{m}\right)}(2^{j_{\ast }}\mathit{x},2^{j_{\ast }}{\mathit{X}}_{\mathit{i}}).\hfill \end{array}$$

Then we can obtain that

$$\begin{array}{c}\hfill \mathbb{P}\left[\left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\right|\ge \kappa {\eta }_n\right]=\mathbb{P}\left[\frac{1}{n}\left|\sum _{i=1}^n{B}_i\right|\ge \kappa {\eta }_n\right].\end{array}$$

(7)

By the definition of $B_i$, $\mathbb{E}\left[B_i\right]=0$. Meanwhile, note that $|K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})|\lesssim 2^{j_{\ast }(d+|\mathit{m}\left|\right)}$ by (5). Now, it follows from A3 and A4 that

$$\begin{array}{c}\hfill |B_i|\lesssim \left|\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right|+\left|\mathbb{E}\left[\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right]\right|\lesssim 2^{j_{\ast }(d+|\mathit{m}\left|\right)}.\end{array}$$

Using the property of the kernel function in (5),

$$\begin{array}{cc}\hfill {\int }_{{[a,b]}^d}{\left(K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{u},\mathit{v})\right)}^{2}d\mathit{v}& =2^{2j_{\ast }(d+|\mathit{m}\left|\right)}{\int }_{{[a,b]}^d}{\left(K^{\left(\mathit{m}\right)}(2^{j_{\ast }}\mathit{u},2^{j_{\ast }}\mathit{v})\right)}^{2}d\mathit{v}\hfill \\ \hfill & =2^{2j_{\ast }(d+|\mathit{m}\left|\right)}{2}^{-j_{\ast }d}{\int }_{{[a,b]}^d}{\left(K^{\left(\mathit{m}\right)}(2^{j_{\ast }}\mathit{u},2^{j_{\ast }}\mathit{v})\right)}^{2}d\left(2^{j_{\ast }}\mathit{v}\right)\hfill \\ \hfill & \lesssim 2^{j_{\ast }(d+2|\mathit{m}\left|\right)}.\hfill \end{array}$$

Then, by A3 and A4, one gets

$$\begin{array}{cc}\hfill \mathbb{E}\left[B_i^2\right]& =Var\left[\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right]\le \mathbb{E}\left[\frac{Y_i^4}{h^2\left({\mathit{X}}_{\mathit{i}}\right)}{\left(K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right)}^2\right]\hfill \\ \hfill & \lesssim \mathbb{E}\left[{\left(K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right)}^2\right]\lesssim {\int }_{{[a,b]}^d}{\left(K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},\mathit{v})\right)}^{2}d\mathit{v}\lesssim 2^{j_{\ast }(d+2|\mathit{m}\left|\right)}.\hfill \end{array}$$

According to the Bernstein’s inequality [26] and the above results, one can obtain that

$$\begin{array}{c}\hfill \mathbb{P}\left[\frac{1}{n}\left|\sum _{i=1}^n{B}_i\right|\ge \kappa {\eta }_n\right]\lesssim exp\left\{-{\displaystyle \frac{n{\kappa }^2{\eta }_n^2}{2\left(2^{j_{\ast }(d+2|\mathit{m}\left|\right)}+\frac{\kappa {\eta }_n{2}^{j_{\ast }(d+|\mathit{m}\left|\right)}}{3}\right)}}\right\}.\end{array}$$

(8)

The conditions ${\eta }_n\asymp {\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}·2^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}$ and $2^{j_{\ast }}\lesssim {\left(\frac{n}{lnn}\right)}^{1/d}$ imply that ${\eta }_n\lesssim 2^{j_{\ast }\left|\mathit{m}\right|}$. Meanwhile, one can easily obtain that

$$\begin{array}{c}\hfill -{\displaystyle \frac{n{\kappa }^2{\eta }_n^2}{2\left(2^{j_{\ast }(d+2|\mathit{m}\left|\right)}+\frac{\kappa {\eta }_n{2}^{j_{\ast }(d+|\mathit{m}\left|\right)}}{3}\right)}}=-{\displaystyle \frac{{\kappa }^2}{2\left(1+\frac{\kappa {\eta }_n}{3·2^{j_{\ast }\left|\mathit{m}\right|}}\right)}}·{\displaystyle \frac{n{\eta }_n^2}{2^{j_{\ast }(d+2|\mathit{m}\left|\right)}}}\lesssim -z\left(\kappa \right)·lnn,\end{array}$$

with $z\left(\kappa \right)=\frac{{\kappa }^2}{2(1+\kappa /3)}$. Then, this result with (7) and (8) implies that

$$\begin{array}{c}\hfill \mathbb{P}\left[\left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\right|\ge \kappa {\eta }_n\right]\lesssim n^{-z\left(\kappa \right)}.\end{array}$$

□

4. Main Theorem

In this section, we will state the convergence rates of the wavelet estimator under different estimation error and mild conditions.

Theorem 1.

For the problem (1), the wavelet estimator $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ is defined by (3), and the following results under different conditions are obtained.

(i)

Let the model (1) satisfy the assumptions A1–A4,

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]|=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\right).\end{array}$$

(9)

(ii)

Assume that the model (1) satisfies the assumptions A1–A4, and the partial derivatives functions $\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)$ belong to Hölder space $H^s\left({\mathbb{R}}^d\right)(s>0)$, one gets

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)|=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}+2^{-j_{\ast }s}\right).\end{array}$$

(10)

(iii)

Suppose that in the model (1) with conditions A3 and A4, the random vector ${\mathit{X}}_{\mathit{1}},\cdots ,{\mathit{X}}_{\mathit{n}}$ is independent. Then, one has

$$\begin{array}{c}\hfill Var\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\lesssim {\displaystyle \frac{2^{j_{\ast }(d+2|\mathit{m}\left|\right)}}{n}}.\end{array}$$

(11)

Remark 1.

If one takes $2^{j_{\ast }}\asymp {\left(\frac{n}{lnn}\right)}^{1/(d+2|\mathit{m}|+2s)}$, then the result of (ii) reduces to

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)|=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{s}{d+2\left|\mathit{m}\right|+2s}}\right).\end{array}$$

Then, when $\mathit{m}=\mathit{0}$, this strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems [27].

Remark 2.

According to Lemma 1, it is easy to know that the wavelet estimator $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ is an unbiased estimator. Hence, the estimation error of this wavelet estimator in the deviation sense is given by (i). In addition, the result (iii) considers the estimation error of the wavelet estimator in the variance sense.

Proof. 

Proof of (i). Note that ${[a,b]}^d$ is a compact set, then it can be covered by a finite number $L_n$ of cubes $I_{\ell }$. Meanwhile, one defines that the centre of $I_{\ell }$ is ${\mathit{x}}_{\ell }:=(x_{{\ell }_1},x_{{\ell }_2},\cdots ,x_{{\ell }_d})$, and the radius length is $l_n:=\frac{c}{{L_n}^{1/d}}$ with a positive constant c. The parametric $L_n$ will be taken in the following discussions. Using the triangle inequality, one gets

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]|\le Q_1+Q_2+Q_3,\end{array}$$

(12)

where

$$\begin{array}{c}{Q}_1:=\underset{1\le \ell \le L_n}{max}\underset{\mathit{x}\in I_{\ell }}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)|,\\ Q_2:=\underset{1\le \ell \le L_n}{max}\underset{\mathit{x}\in I_{\ell }}{sup}|\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)\right]-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]|,\\ Q_3:=\underset{1\le \ell \le L_n}{max}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)\right]|.\end{array}$$

For $Q_1$. By the definitions of $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ and ${\widehat{\alpha }}_{j_{\ast },\mathit{k}}$ in (3) and (4), for any $\mathit{x}\in {[a,b]}^d$, one can easily obtain

$$\begin{array}{cc}\hfill & |\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)|\hfill \\ \hfill & =\left|\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}{\widehat{\alpha }}_{j_{\ast },\mathit{k}}({\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)-{\Phi }_{j_{\ast },\mathit{k}}\left({\mathit{x}}_{\ell }\right))\right|\hfill \\ \hfill & \le \frac{1}{n}\left|\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}\left[K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})-K_{j_{\ast }}^{\left(\mathit{m}\right)}({\mathit{x}}_{\ell },{\mathit{X}}_{\mathit{i}})\right]\right|\hfill \\ \hfill & +\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}\left|{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}·({\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)-{\Phi }_{j_{\ast },\mathit{k}}\left({\mathit{x}}_{\ell }\right))\right|\hfill \\ \hfill & =:Q_{11}+Q_{12}.\hfill \end{array}$$

(13)

Using A3, A4 and (6),

$$\begin{array}{c}\hfill Q_{11}\lesssim \frac{1}{n}\left|\sum _{i=1}^n\left[K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})-K_{j_{\ast }}^{\left(\mathit{m}\right)}({\mathit{x}}_{\ell },{\mathit{X}}_{\mathit{i}})\right]\right|\lesssim 2^{j_{\ast }(d+1+|\mathit{m}\left|\right)}\parallel \mathit{x}-{\mathit{x}}_{\ell }\parallel .\end{array}$$

(14)

In addition, by the boundedness assumption of function $g\left(\mathit{x}\right)$ in A2, the following inequalities are true:

$$\begin{array}{cc}\hfill & \left|{\int }_{{[a,b]}^d}{g}^2\left(\mathit{x}\right)\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}\right|\hfill \\ \hfill & \lesssim \left|{\int }_{{[a,b]}^d}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left(\mathit{x}\right)d\mathit{x}\right|\hfill \\ \hfill & =2^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\left|{\int }_{{[a,b]}^d}\left({\partial }^{\mathit{m}}\Phi \right)(2^{j_{\ast }}{x}_1-k_1,\cdots ,2^{j_{\ast }}{x}_d-k_d)d\mathit{x}\right|\hfill \\ \hfill & \le 2^{j_{\ast }(-\frac{d}{2}+\left|\mathit{m}\right|)}{\int }_{{[a,b]}^d}\left|\left({\partial }^{\mathit{m}}\Phi \right)(2^{j_{\ast }}{x}_1-k_1,\cdots ,2^{j_{\ast }}{x}_d-k_d)\right|d(2^{j_{\ast }}\mathit{x}-\mathit{k})\hfill \\ \hfill & \lesssim 2^{j_{\ast }(-\frac{d}{2}+\left|\mathit{m}\right|)}.\hfill \end{array}$$

Furthermore, it follows from the property of wavelet scaling function $\Phi$ that

$$\begin{array}{cc}\hfill Q_{12}& \lesssim 2^{j_{\ast }(-\frac{d}{2}+\left|\mathit{m}\right|)}\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}\left|{\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)-{\Phi }_{j_{\ast },\mathit{k}}\left({\mathit{x}}_{\ell }\right)\right|\hfill \\ \hfill & =2^{j_{\ast }(-\frac{d}{2}+\left|\mathit{m}\right|)}·2^{\frac{j_{\ast }d}{2}}\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}\left|\Phi (2^{j_{\ast }}\mathit{x}-\mathit{k})-\Phi (2^{j_{\ast }}{\mathit{x}}_{\ell }-\mathit{k})\right|\hfill \\ \hfill & =2^{j_{\ast }\left|\mathit{m}\right|}\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}\left|▽\Phi (2^{j_{\ast }}\mathit{x}-\mathit{k})·{(2^{j_{\ast }}{x}_1-2^{j_{\ast }}{x}_{{\ell }_1},\cdots ,2^{j_{\ast }}{x}_d-2^{j_{\ast }}{x}_{{\ell }_d})}^T\right|\hfill \\ \hfill & \lesssim 2^{j_{\ast }(1+|\mathit{m}\left|\right)}\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}∥\mathit{x}-{\mathit{x}}_{\ell }∥\lesssim 2^{j_{\ast }(d+1+|\mathit{m}\left|\right)}∥\mathit{x}-{\mathit{x}}_{\ell }∥.\hfill \end{array}$$

(15)

Combining (13), (14) and (15), one can get

$$\begin{array}{c}\hfill |\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)|\lesssim 2^{j_{\ast }(d+1+|\mathit{m}\left|\right)}∥\mathit{x}-{\mathit{x}}_{\ell }∥.\end{array}$$

Because the centre of $I_{\ell }$ is ${\mathit{x}}_{\ell }$, $∥\mathit{x}-{\mathit{x}}_{\ell }∥\lesssim l_n$. Then, by the definition of $l_n$,

$$\begin{array}{c}\hfill Q_1\lesssim 2^{j_{\ast }(d+1+|\mathit{m}\left|\right)}{l}_n\lesssim \frac{2^{j_{\ast }(d+1+|\mathit{m}\left|\right)}}{{L_n}^{1/d}}.\end{array}$$

Now, one takes

$$\begin{array}{c}\hfill L_n\asymp {\left(\frac{2^{j_{\ast }(d+2)}n}{lnn}\right)}^{\frac{d}{2}}.\end{array}$$

Then, the following conclusion is true,

$$\begin{array}{c}\hfill Q_1\lesssim {\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}.\end{array}$$

(16)

For $Q_2$. Using the above discussions of $Q_1$, one knows

$$\begin{array}{c}\hfill Q_2\le \underset{1\le \ell \le L_n}{max}\underset{\mathit{x}\in I_{\ell }}{sup}\mathbb{E}\left[\right|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)-\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\left|\right]\lesssim {\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}.\end{array}$$

(17)

For $Q_3$. Note that

$$\begin{array}{cc}\hfill \mathbb{P}\left[Q_3\ge \kappa {\eta }_n\right]& =\mathbb{P}\left[\underset{1\le \ell \le L_n}{max}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)\right]|\ge \kappa {\eta }_n\right]\hfill \\ \hfill & \le \sum _{\ell =1}^{L_n}\mathbb{P}\left[|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left({\mathit{x}}_{\ell }\right)\right]|\ge \kappa {\eta }_n\right]\hfill \\ \hfill & \le L_n\underset{\mathit{x}\in {[a,b]}^d}{sup}\mathbb{P}\left[|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]|\ge \kappa {\eta }_n\right].\hfill \end{array}$$

By Lemma 2 and $2^{j_{\ast }}\lesssim {\left(\frac{n}{lnn}\right)}^{1/d}$, one can choose a large enough constant $\kappa$ such that

$$\begin{array}{c}\hfill \sum _{i=1}^n\mathbb{P}\left[Q_3\ge \kappa {\eta }_n\right]\lesssim \sum _{i=1}^n{L}_n{n}^{-z\left(\kappa \right)}<\infty .\end{array}$$

Furthermore, this result with the Borel–Cantelli lemma implies

$$\begin{array}{c}\hfill Q_3=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}·2^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\right).\end{array}$$

(18)

Finally, together with (12), (16), (17), and (18), one gets

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]|=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\right).\end{array}$$

(19)

Proof of (ii). Using Lemma 1 and the property (2) of wavelets,

$$\begin{array}{cc}\hfill \left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)\right|& \le \left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\right|+\left|\sum _{j=j_{\ast }}^{\infty }\sum _{u=1}^{2^d-1}\sum _{\mathit{k}\in {\Lambda }_j}{\beta }_{j,\mathit{k},u}{\Psi }_{j,\mathit{k},u}\left(\mathit{x}\right)\right|.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}\left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)\right|\le I_1+I_2,\end{array}$$

(20)

where

$$\begin{array}{c}{I}_1:=\underset{\mathit{x}\in {[a,b]}^d}{sup}\left|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\mathbb{E}\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\right|,\\ I_2:=\underset{\mathit{x}\in {[a,b]}^d}{sup}\left|\sum _{j=j_{\ast }}^{\infty }\sum _{u=1}^{2^d-1}\sum _{\mathit{k}\in {\Lambda }_j}{\beta }_{j,\mathit{k},u}{\Psi }_{j,\mathit{k},u}\left(\mathit{x}\right)\right|.\end{array}$$

For $I_1$. According to the conclusion of (i),

$$\begin{array}{c}\hfill I_1=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}\right).\end{array}$$

(21)

For $I_2$. Let a function $f\left(\mathit{x}\right)$ belong to Hölder space $H^s\left({\mathbb{R}}^d\right)$, and let ${\Psi }_{j,\mathit{k},u}$ be a wavelet function, then one can prove that $\left|{\displaystyle \sum _{j=j_{\ast }}^{\infty }\sum _{u=1}^{2^d-1}\sum _{\mathit{k}\in {\Lambda }_j}}{\beta }_{j,\mathit{k},u}{\Psi }_{j,\mathit{k},u}\left(\mathit{x}\right)\right|\lesssim 2^{-j_{\ast }s}.$ More details and proofs of this above conclusion can be found in [28,29,30]. Furthermore, because the partial derivatives functions $\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)$ belong to Hölder space $H^s\left({\mathbb{R}}^d\right)$, one can easily obtain that

$$I_2\lesssim 2^{-j_{\ast }s}.$$

Now, this conclusion with (20) and (21) shows that

$$\begin{array}{c}\hfill \underset{\mathit{x}\in {[a,b]}^d}{sup}|\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)-\left({\partial }^{\mathit{m}}r\right)\left(\mathit{x}\right)|=O_{a.s.}\left({\left(\frac{lnn}{n}\right)}^{\frac{1}{2}}{2}^{j_{\ast }(\frac{d}{2}+\left|\mathit{m}\right|)}+2^{-j_{\ast }s}\right).\end{array}$$

Proof of (iii). By the definition of $\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)$ and the properties of the variance function, one has

$$\begin{array}{cc}\hfill Var\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]& =Var\left[\sum _{\mathit{k}\in {\Lambda }_{j_{\ast }}}\left(\frac{{(-1)}^{\left|\mathit{m}\right|}}{n}\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}\left({\partial }^{\mathit{m}}{\Phi }_{j_{\ast },\mathit{k}}\right)\left({\mathit{X}}_{\mathit{i}}\right)\right){\Phi }_{j_{\ast },\mathit{k}}\left(\mathit{x}\right)\right]\hfill \\ \hfill & =Var\left[\frac{{(-1)}^{\left|\mathit{m}\right|}}{n}\sum _{i=1}^n\frac{Y_i^2}{h\left({\mathit{X}}_{\mathit{i}}\right)}{K}_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{i}})\right].\hfill \end{array}$$

Moreover, the assumptions of random vector $\mathit{X}$, A3 and A4 imply that

$$Var\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\lesssim \frac{1}{n}Var\left[K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{1}})\right].$$

Using the property of the kernel function in (5) and condition A3,

$$\begin{array}{cc}\hfill Var\left[K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{1}})\right]& \le \mathbb{E}\left[{\left(K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},{\mathit{X}}_{\mathit{1}})\right)}^2\right]\hfill \\ \hfill & \lesssim {\int }_{{[a,b]}^d}{\left(K_{j_{\ast }}^{\left(\mathit{m}\right)}(\mathit{x},\mathit{v})\right)}^{2}d\mathit{v}\lesssim 2^{j_{\ast }(d+2|\mathit{m}\left|\right)}.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill Var\left[\widehat{\left({\partial }^{\mathit{m}}r\right)}\left(\mathit{x}\right)\right]\lesssim \frac{2^{j_{\ast }(d+2|\mathit{m}\left|\right)}}{n}.\end{array}$$

□

5. Conclusions

For nonparametric derivative estimation, classical research results pay more attention to the derivative estimation of one-dimensional functions. However, this paper studies the nonparametric estimation of partial derivatives of a multivariate function. Firstly, a wavelet estimator of the partial derivatives of the multivariate variance function in a heteroscedastic model is given. More importantly, this wavelet estimator is an unbiased estimator. Secondly, two important lemma are proved, which discuss the key properties of the wavelet estimator. Finally, the convergence rates over different estimation errors of the wavelet estimator are considered. According to the main theorem, it is easy to see that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function estimations.

Because the local analysis characteristics in the time and frequency domains of the wavelet, the wavelet estimator can choose an appropriate wavelet scaling parameter to get the optimal convergence rate. Hence, this paper considers partial derivatives estimation based on the wavelet method. The theoretical results of asymptotic property of the wavelet estimator are discussed in this paper. In addition, it is difficult to present the corresponding practical illustration of the wavelet estimator, which needs more investigations and some new skills. We will consider this in future work.